
Intuitively, for every transmission vector \({\mathbf {x}}_{\hat{{\mathcal {K}}}}\)  we should have \(\frac{|Y_{{\mathcal {U}},k}|}{R_k} = \frac{|Y_{{\mathcal {U}},i}|}{R_i}\) , \(\forall k,i \in \hat{{\mathcal {K}}}\) , where \(R_k\)  is the dedicated transmission rate to user \(k\)  (detailed discussions are left for the extended version of this paper). This simply results in \(\frac{R_k}{1-m_k} = \frac{R_i}{1-m_i}\) , \(\forall k,i \in \hat{{\mathcal {K}}}\) . Let us define \(R_w = \frac{R_k}{1-m_k}\)  as the common weighted rate. Then, the transmission time for vector \({\mathbf {x}}_{\hat{{\mathcal {K}}}}\)  would be \(T_{\hat{{\mathcal {K}}}} = {K}/{\binom{K}{\hat{t}+L}\binom{\hat{t}+L}{\hat{t}}LR_w}\) , which is independent of users' states. Now, assuming \(R_w\)  is almost the same for any subset of users \(\hat{{\mathcal {K}}}\) , the total delivery time can be approximated as \(T_{T} \approx {K}/{\binom{\hat{t}+L}{\hat{t}}LR_w}\) , and the symmetric rate will be \(R_w^s = \frac{K}{T_T} = {\binom{\hat{t}+L}{\hat{t}}LR_w}\) . Following a similar argument for the multiantenna CC scheme in [1]}, the symmetric rate would be \(R_u^s = \frac{K}{T_T} = {\binom{t+L}{t}LR_u}\) , where \(R_u = \frac{\bar{R}}{1-M/S}\) , \(\bar{R}\)  is the common max-min rate for transmitting every data element, and \(t = {KM}/{S}\)  is the common caching gain (c.f. [1]} Section IV).
